Finite Group Databasebeta

← C4×#2228 #495978

Subgroups — C4×#2228 #495978

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#13277C211yesmaximal, characteristic, complemented
5#13178C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#12937C211yesmaximal, characteristic
9#12935C211yesmaximal, characteristic
10#12813C211yesmaximal, characteristic
11C4×#367C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13C4×C4⋊C16C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C4×#125—21no—
21C42×D4—21no—
22#2228C418yesdirect product factor
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1991—21no—
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1941—21no—
39#1846C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#106—21no—
46C4×#102—21no—
47C4×#102—21no—
48C4×#102—21no—
49C42×C8C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#125—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C22×C42—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#499—22no—
71#499C2×C414yescomplemented
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8—22no—
76C4×D8C2×C414yes—
77C4×D8Q814yes—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#389—21no—
97#389—21no—
98#387—41no—
99#387—41no—
100#382—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—41no—
107C4×C4⋊C4—41no—
108C4×C22⋊C4—41no—
109C4×C22⋊C4—41no—
110C4×C22⋊C4—41no—
111C43D411yescharacteristic
112#367—22no—
113#367C2×C414yescomplemented
114#366—22no—
115#366C2×C414yescomplemented
116C4⋊C16—22no—
117C4⋊C16C2×C414yescomplemented
118#357—22no—
119#357—22no—
120#357C2×C414yes—
121#357C2×C414yes—
122C4×C16—22no—
123C4×C16—22no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131C22×D4—21no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8C22×C412yes—
139C2×D8C2×Q812yes—
140C2×D8#4112yes—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2411yescharacteristic
145#125—24no—
146#125—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—84no—
173C2×C42D811yescharacteristic
174C2×C42D811yescharacteristic
175C2×C42D811yescharacteristic
176C2×C42D811yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16C22×C412yes—
191C2×C16C22×C412yes—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107C22×C412yes—
202#107C22×C412yes—
203#107C22×C412yes—
204#107C2×Q812yes—
205#107#4112yes—
206#106—24no—
207#106—42no—
208#102—24no—
209#102—24no—
210#102—24no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—41no—
216C4×C8C22×C412yes—
217C4×C8C2×Q812yes—
218C4×C8#4112yes—
219C4×C8#4112yes—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229C24—41no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C22×C4—44no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C22×D411yescharacteristic
259D8—22no—
260D8—22no—
261D8—22no—
262D8C4×Q814yes—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8C4×D412yes—
271C2×C8C22⋊Q812yes—
272C2×C8C23×C411yescharacteristic
273C2×C8C22×Q811yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—82no—
306C42—82no—
307C42—82no—
308C42—82no—
309C42—82no—
310C42C4×D412yes—
311C42C22⋊Q812yes—
312C42#13512yes—
313C42#13712yes—
314C16—22no—
315C16—22no—
316C16—41no—
317C16—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23C22×D811yescharacteristic
326D4—24no—
327D4—42no—
328D4—42no—
329D4—42no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—88no—
347C2×C4—88no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—88no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4C4×D812yes—
366C2×C4C4×D812yes—
367C2×C4#47412yes—
368C2×C4#47412yes—
369C2×C4#50812yes—
370C2×C4#50812yes—
371C2×C4#50912yes—
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4#52511yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4#57511yescharacteristic
378C8—22no—
379C8—22no—
380C8—41no—
381C8C2×C4×Q811yescharacteristic
382C8C4×#4111yescharacteristic
383C8#52911yescharacteristic
384C8#53311yescharacteristic
385C22—44no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22C2×C4×D811yescharacteristic
393C22C2×#47411yescharacteristic
394C22#310011yescharacteristic
395C22C2×#50811yescharacteristic
396C22C2×#50811yescharacteristic
397C22C2×#50911yescharacteristic
398C22#341411yescharacteristic
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—41no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—82no—
409C4—161no—
410C4—161no—
411C4#222814yesdirect product factor
412C4C4×C22⋊Q811yescharacteristic
413C4C4×#13711yescharacteristic
414C4#307611yescharacteristic
415C4#310711yescharacteristic
416C2—84no—
417C2C4×#47411yescharacteristic
418C2C4×#50811yescharacteristic
419C2C4×#50911yescharacteristic
420C2C2×#222811yescharacteristic
421C2#3362611yescharacteristic
422C2#3366211yescharacteristic
423C2#3366511yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4×#499C2semidirect
4. #13277C2semidirect
5. #13178C2semidirect
6. #13112C2semidirect
7. #13110C2semidirect
11. C4×#367C2semidirect
12. C4×#366C2semidirect
13. C4×C4⋊C16C2semidirect
22. #2228C4direct (N × H)
71. #499C2×C4semidirect
113. #367C2×C4semidirect
115. #366C2×C4semidirect
117. C4⋊C16C2×C4semidirect
411. C4#2228direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22282
C4×#4992
C42×D82
#132772
#131782
#131122
#131102
#129372
#129352
#128132
C4×#3672
C4×#3662
C4×C4⋊C162
C4×#3572